3.3.2 Problem formulation of stage (II)

In Stage (II), the objective is to minimize the DG active power curtailment, where the final SOC reached in Stage (I), that is, SOCR chð Þ<sup>i</sup> ð Þ;<sup>t</sup> , must be attained to ensure maximum customer satisfaction, which is the highest priority of the V2GQ technology. Therefore, this stage is subject to all of the constraints in Stage (I) except for (47), which is replaced by

$$\text{SOC}^{\text{F}}\_{\left(ch\_{(i)},t\right)} = \text{SOC}^{\text{R}}\_{\left(ch\_{(i)},t\right)}, \quad \forall i \in \mathcal{T}\_{\text{PEV}}, ch\_{(i)}, t \tag{48}$$

Thus, the objective function of Stage (II) is

$$\max\_{\mathcal{T}} \sum\_{i \in \mathcal{T}\_{DG}} P\_{o(i,t)}, \qquad \forall t \tag{49}$$

#### 3.3.3 Problem formulation of stage (III)

Stage (III) aims at minimizing the voltage deviation using the DGs and PEVs to restore a feasible solution for the COC and relax the tap operation. Thus,

$$\min\_{\mathcal{V}} \sum\_{i \in \mathcal{T}\_b} \left( \mathbf{1} - \mathbf{V}\_{(i,t)} \right)^2, \quad \forall i, t \tag{50}$$

Besides all the constraints in Stage (II), this problem is subject to the constraint defined in (50), in which the maximum injected powers from the DGs reached in Stage (II), that is, P<sup>R</sup> o ið Þ ;<sup>t</sup> , should be maintained.

$$P\_{o(i,t)} = P\_{o(i,t)}^{\mathbb{R}} \quad \forall i \in \mathcal{T}\_{DG}, t \tag{51}$$

#### 3.4 Coordination between V2GQ and COC

The charging decisions and active/reactive dispatch signals produced in Stage (III) are sent to all PEV parking lots and DGs, as shown in Figure 9. To ensure that the PEV and DG converters settle at the desired active and reactive power references, a time delay Δtconv is introduced. The converter settling time may vary from 50 to 100 ms, depending on the primary controllers of the DC/AC converter [7]. For slow automatic interactions, such as voltage regulation, the maximum communication time delay is 100 ms as per the IEC 61850 [20]. Thus, Δtconv is assumed to be
